MAT 342, Applied Complex Analysis

Fall 2016

Christopher Bishop

Professor, Mathematics
SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631
bishop at

Time and place: MWF 10:00-10:53am, MATH P-131 (ROOM CHANGED from Physics P-112)

academic calendar

Exam calendar

Grader: Jack Burkart, jack.burkart at, office hours Wed 11am-12pm in 5-125b (Math Tower); TuTh 1pm-2pm in Math Learning Center (S-level = basement in Math Tower)

FINAL EXAM : Tuesday, December 20, 2:15-5:00pm in usual room (P-131 Math Building).

REVIEW SESSION : Friday, December 16, 10-11am in usual room (P-131 Math Building).

practice first midterm This is 50 questions. The actual midterm will have exactly the same format, although the questions will be changed. Most of the questions are not hard, but you will have to move quickly to finish the whole exam. We will go over the practice in class on Monday, Oct 10. The midterm is intended to cover Chapters 1-4. A few questions on the midterm refer to things that we have not yet covered in class, but will cover before the midterm (if we don't, then I will make sure this material in not on the actual midterm).

answers for practice first midterm

answers for first midterm

histogram of results for first midterm

practice second midterm The actual midterm is Wednesday, October 16, and will look simliar to the practice, but may contain a few questions with a different format. The practice exam was written quickly and may have some typos. I will try to post an answer key by Sunday (Oct 13) evening.

answers for practice second midterm

answers for second midterm

histogram of results for second midterm

Final Exam: Tuesday, December 20, 2016 2:15pm-5:00pm, location to be announced (probably during last week of classes)

The textbook "Complex Variables and Applications" by J.W. Brown and R.V. Churchill, 9th edition. The maerial covered in earlier editions should be similar, but sections and exericses might be numbered differently. All assignments will be based on 9th edition, so if you are using a different edition, it is your responsibility to make sure you do the correct work. 8th edition

This is an introduction to functions of a complex variable and emphasizes developing computational skill with complex numbers (complex arithmetic, power series manipulation, evaluation of real and complex integrals using residues,...). It is also a mathematically rigorous course, and most statements will come with complete proofs. Students will be expected to be able to do simple proofs and derivations, as well as perform the calculational skills mentioned above. Topics covered will include properties of complex numbers, analytic functions with examples, contour integrals, the Cauchy integral formula, the fundamental theorem of algebra, power series and Laurant series, residues and poles with applications, conformal mappings with applications and other topics if time permits.

An alternative (or sequel) to this course is MAT 536 (previously numbered MAT 542). This is a first year graduate course in one complex variable that is offered every Spring. It covers about twice as much material at MAT 342 and is much more theoretical (all proofs, all the time).

Grades will be based on weekly problem sets (50 points total), two midterms (50 points each) and a final exam (50 points).

Prof. Bishop's Office Hours: MW 9-10, M 2-3. You should aso feel free to make appointments for other times, or just stop by my office.

tentative lecture schedule and problem assignments
The schedule may change as we proceed, and this may change the assigments. I will post updated copies here if that happens.

I updated the homework on Sept 23,to correct a type in the assignment for Sept 26. Problems that were marked as in Section 37 did not exist and were meant to be in Section 36.

Send me email at: bishop at (I usually respond faster to this than to my account)

Send the grader an email at: jack.burkart at

What we did in class:

Mon. Aug 29: introduction, the complex plane, complex addition and mulitplication, some advantages of using complex numbers, the field axioms
Wed. Aug 31: review addition and multiplication, real and imginary parts, complex conjugates, absolute value, division, polar coordiates, unit complex numbers, powers of i, triangle inequality, squaring , square roots
Fri. Sept 2: exponential form (with motivation based on power series), Euler's formula, find powers and roots, de Moivre's formula, the quadratic formula. Introduction to planar topology, balls, open and closed sets, boundary of a set, interior and exterior points, connected set, bounded versus unbounded, accumulation point.
Wed. Sept 7: Review planar topology, functions, domains, real-valued, polynomials, rational functions, image inverse image, translation, rotation, dilation, reflection, geometryof z^2, definition of limit, uniqueness, real and imaginary parts, addition of limits, multiplication of limits, limits involving infinity.
Fri. Sept 9: two definitions of continuity, composition of continuous functions, real and imaginary parts of continuous functions, boundedness, definition of derivative, examples, rules for differentiation.
Mon. Sept 12: Cauchy-Riemann equations, differentiable implies CR euations hold, examples, CR equations imply differentiable.
Wed. Sept 14: Sections 24,25,26. Polar form of Cauchy-Riemann equations, definition and examples of analytic functions, zero deriviative implies constant function.
Fri. Sept 16 : Sections 27, 28,29. Harmonic functions, uniquely determined analytic functions, the reflection principle.
Mon. Sept 19: Sections 30, 31, 32, 33. Exponentials and logarithms.
Wed. Sept 21: Sections 34, 35, 36, 37. More about logs. Power functions, trigonometric functions.
Fri. Sept 23 : Sections 38, 39, 40. Inverse trig functions, hyperbolic functions
Mon. Sept 26: Sections 41, 42, 43. Derivatives, integrals, contours.
Wed. Sept 28: Sections 44, 45, 46, 47. Contour integrals, examples, branch cuts, upper bounds.
Fri. Sept 30 : Sections 48, 49 Independence of integral from contour; statement and proof.
Mon. Oct 3: Sections 50, 51. Cauchy-Goursat theorem; statement and proof.
Wed. Oct. 5: Sections 52, 53. Simply connected and multiply connected domains.
Fri. Oct 7: Liouville's theorem, fundamental theorem of algebra,
Mon. Oct 10: Review practice Midterm
Wed. Oct. 12: Maximum principle, review for Midterm
Fri. Oct 14: Midterm
Mon. Oct 17: Introduction to power series, Taylor series for analytic functions
Wed. Oct. 19: Laurent series
Fri. Oct 21: Uniform convergence, continuity of limit, limits of integrals

Mon. Oct 24: Uniform convergence of power series, uniqueness, term-by-term integration and differentiation, multiplication and division of series.
Wed. Oct 26: Isolated singularities, residues, Cauchy's residue theorem
Fri. Oct 28: Residue at infinity, three types of point singularities, examples
Mon. Oct 31, go over first midterm :
Wed. Nov 2: Residues at poles, examples
Fri. Nov 4: zeros of analytic functions, zeros and poles,
Mon, Nov 7 : removable singularities, essential singularities and poles
Wed. Nov 9 Evaluation of integrals using residues
Fri. Nov 11: Evaluation of integrals using residues
Mon, Nov 14 : review for Midterm 2
Wed. Nov 16 Midterm 2
Fri. Nov 18: Argument Principle, Rouche's theorem

What I plan to do next

Mon. Nov 21: Class canceled. Problems due today will be due Nov 28 instead.
Mon. Oct 28: Linear and linear fractional transformations, applications to topology of surfaces and manifolds
Wed Nov 30: go over second midterm
Fri. Dec 2. Guest Lecture by Prof Mulhaupt of AMS: complex analysis and quantitative finance.
Mon. Dec 5: Guest Lecture by Prof Sutherland: root finding and fractals
Wed. Dec 7: Conformal mapping
Fri. Dec 9 Review for final.
Tue. Dec 20 final exam, 2:15-5:00pm

Additional links

Link to history of mathematics